Conceptual Issues in Inflation

New Inflation

1981 - 1982

V

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Chaotic Inflation

1983

Eternal Inflation

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Hybrid Inflation

1991, 1994

WMAP5 Acbar Boomerang CBI

Tensor modes

Kallosh, A.L. 2007

It does make sense to look for tensor modes even

if none are found at the level r 0.1 (Planck)

Blue lines chaotic inflation with the simplest

spontaneous symmetry breaking potential

for N 50 and N 60

Destri, de Vega, Sanchez, 2007

Possible values of r and ns for chaotic

inflation with a potential including terms

for N 50. The color-filled

areas correspond to various confidence

levels according to the WMAP3 and SDSS data.

Almost all points in this area can be fit by

chaotic inflation including terms

What is fNL?

Komatsu 2008

k2

k1

k3

- fNL the amplitude of three-point function
- also known as the bispectrum, B(k1,k2,k3),

which is - ltF(k1)F(k2)F(k3)gtfNL(2p)3d3(k1k2k3)b(k1,k2,k3)
- F(k) is the Fourier transform of the

curvature perturbation, and b(k1,k2,k3) is a

model-dependent function that defines the shape

of triangles predicted by various models.

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Why Bispectrum?

- The bispectrum vanishes for Gaussian random

fluctuations. - Any non-zero detection of the bispectrum

indicates the presence of (some kind of)

non-Gaussianity. - A very sensitive tool for finding non-Gaussianity.

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Two fNLs

Komatsu Spergel (2001) Babich, Creminelli

Zaldarriaga (2004)

- Depending upon the shape of triangles, one can

define various fNLs - Local form
- which generates non-Gaussianity locally (i.e., at

the same location) via F(x)Fgaus(x)fNLlocalFgau

s(x)2 - Equilateral form
- which generates non-Gaussianity in a different

way (e.g., k-inflation, DBI inflation)

Earlier work on the local form SalopekBond

(1990) Gangui et al. (1994) Verde et al.

(2000) WangKamionkowski (2000)

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Can we have large nongaussianity ?

A.L., Kofman 1985-1987, A.L., Mukhanov,

1996, Lyth, Wands, Ungarelli, 2002 Lyth, Wands,

Sasaki and collaborators - many papers up to 2008

V

V

?

- Inflaton

Curvaton

Isocurvature perturbations

adiabatic perturbations

is determined by quantum

fluctuations, so the amplitude of perturbations

is different in different places

?

Spatial Distribution of the Curvaton Field

?

0

The Curvaton Web and Nongaussianity

Usually we assume that the amplitude of

inflationary perturbations is constant, ?H

10-5 everywhere. However, in the curvaton

scenario ?H can be different in different parts

of the universe. This is a clear sign of

nongaussianity.

A.L., Mukhanov, astro-ph/0511736

The Curvaton Web

?H

Alternatives? Ekpyrotic/cyclic scenario

Original version (Khoury, Ovrut, Steinhardt and

Turok 2001) did not work (no explanation of the

large size, mass and entropy the homogeneity

problem even worse than in the standard Big Bang,

Big Crunch instead of the Big Bang, etc.).

It was replaced by cyclic scenario (Steinhardt

and Turok 2002) which is based on a set of

conjectures about what happens when the universe

goes through the singularity and re-emerges.

Despite many optimistic announcements, the

singularity problem in 4-dimensional space-time

and several other problems of the cyclic scenario

remain unsolved.

Recent developments New ekpyrotic scenario

Creminelly and Senatore, 2007, Buchbinder,

Khoury, Ovrut 2007

Problems violation of the null energy condition,

absence of the ultraviolet completion, difficulty

to embed it in string theory, violation of the

second law of thermodynamics, problems with black

hole physics.

The main problem this theory contains terms

with higher derivatives, which lead to new

ekpyrotic ghosts, particles with negative energy.

As a result, the vacuum state of the new

ekpyrotic scenario suffers from a catastrophic

vacuum instability.

Kallosh, Kang, Linde and Mukhanov, arXiv0712.2040

The New Ekpyrotic Ghosts

New Ekpyrotic Lagrangian

Dispersion relation

Two classes of solutions, for small P,X

,

Hamiltonian describes normal particles with

positive energy ?1 and ekpyrotic ghosts with

negative energy - ?2

Vacuum in the new ekpyrotic scenario instantly

decays due to emission of pairs of ghosts and

normal particles.

Cline, Jeon and Moore, 2003

Why higher derivatives? Can we introduce a UV

cutoff?

Bouncing from the singularity requires violation

of the null energy condition, which in turn

requires

Dispersion relation for perturbations of the

scalar field

The last term appears because of the higher

derivatives. If one makes this term vanish at

large k, then in the regime

one has a catastrophic vacuum instability, with

perturbations growing as

Of course, one can simply assume the existence of

a UV cutoff at energies higher than the ghost

mass, but this would be an inconsistent theory,

until the physical origin of the cutoff is found.

Moreover, in a theory with a UV cutoff, why would

one care about the cosmological singularity? At

this level, the singularity problem could be

solved many decades ago (no high energy modes

no space-time singularity).

But ghosts have disastrous consequences for the

viability of the theory. In order to regulate the

rate of vacuum decay one must invoke explicit

Lorentz breaking at some low scale. In any case

there is no sense in which a theory with ghosts

can be thought as an effective theory, since the

ghost instability is present all the way to the

UV cut-off of the theory.

Buchbinder, Khoury,

Ovrut 2007

- Can we save this theory?

Example

Can be obtained by integration with respect to

of the theory with ghosts

By adding some other terms and integrating out

the field one can reduce this theory to

the ghost-free theory.

Creminelli, Nicolis, Papucci and Trincherini, 2005

But this can be done only for a 1, whereas in

the new ekpyrotic scenario a - 1

Kallosh, Kang, Linde and Mukhanov, arXiv0712.2040

Even if it is possible to improve the new

ekpyrotic scenario (which was never

demonstrated), then it will be necessary to check

whether the null energy condition is still

violated in the improved theory despite the

postulated absence of ghosts. Indeed, if the

correction will also correct the null energy

condition, then the bounce will become

impossible. We are unaware of any examples of

the ghost-free theories where the null energy

condition is violated.

Other alternatives String gas cosmology

Brandenberger, Vafa, Nayeri, 4 papers in 2005-2006

Many loose ends and unproven assumptions (e.g.

stabilization of the dilaton and of extra

dimensions). Flatness/entropy problem is not

solved. This class of models differs from the

class of stringy models where stabilization of

all moduli was achieved.

Even if one ignores all of these issues, the

perturbations generated in these models are very

non-flat Instead of ns 1 one

finds ns 5

Kaloper, Kofman, Linde, Mukhanov 2006,

hep-th/0608200 Brandenberger et al, 2006

Other problems with the string gas constructions

were recently discussed by Kaloper and Watson,

arXiv0712.1820

A toy model of SUGRA inflation

Holman, Ramond, Ross, 1984

Superpotential

Kahler potential

Inflation occurs for ?0 1 Requires

fine-tuning, but it is simple, and it works

A toy model of string inflation

A.L., Westphal, 2007

Superpotential

Kahler potential

Volume modulus inflation Requires fine-tuning,

but works without any need to study complicated

brane dynamics

String Cosmology and the Gravitino Mass

Kallosh, A.L. 2004

The height of the KKLT barrier is smaller than

VAdS m23/2. The inflationary potential Vinfl

cannot be much higher than the height of the

barrier. Inflationary Hubble constant is given by

H2 Vinfl/3 lt m23/2.

uplifting

Modification of V at large H

VAdS

Constraint on the Hubble constant in this class

of models

H lt m3/2

Can we avoid these conclusions?

Recent model of chaotic inflation is string

theory (Silverstein and Westphal, 2007) also

require .

H lt m3/2

In more complicated theories one can have

. But this

requires fine-tuning (Kallosh, A.L. 2004,

Badziak, Olechowski, 2007)

In models with large volume of compactification

(Quevedo et al) the situation is even more

dangerous

It is possible to solve this problem, but it is

rather nontrivial.

Conlon, Kallosh, A.L., Quevedo, in preparation

Remember that we are suffering from the light

gravitino and the cosmological moduli problem for

the last 25 years.

The price for the SUSY solution of the hierarchy

problem is high, and it is growing. Split

supersymmetry? We are waiting for LHC...

Tensor Modes and GRAVITINO

Kallosh, A.L. 2007

superheavy gravitino

unobservable

A discovery or non-discovery of tensor modes

would be a crucial test for string theory and

particle phenomenology

Landscape of eternal inflation

Perhaps 101000 different uplifted vacua

Lerche, Lust, Schellekens 1987

Bousso, Polchinski 2000 Susskind 2003 Douglas,

Denef 2003

What is so special about our world?

Problem Eternal inflation creates infinitely

many different parts of the universe, so we must

compare infinities

Two different approaches

- Study events at a given point, ignoring growth of

volume, or, equivalently, calculating volume in

comoving coordinates - Starobinsky 1986, Garriga,

Vilenkin 1998, Bousso 2006, A.L. 2006

No problems with infinities, but the results

depend on initial conditions. It is not clear

whether these methods are appropriate for

description of eternal inflation, where the

exponential growth of volume is crucial.

2. Take into account growth of volume

A.L. 1986 A.L., D.Linde, Mezhlumian,

Garcia-Bellido 1994 Garriga, Schwarz-Perlov,

Vilenkin, Winitzki 2005 A.L. 2007

No dependence on initial conditions, but we are

still learning how to do it properly.

Let us discuss non-eternal inflation to learn

about the measure

The universe is divided into two parts, one

inflates for a long time, one does it for a short

time. Both parts later collapse.

More observers live in the inflationary (part of

the) universe because there are more stars and

galaxies there

Comoving probability measure does not distinguish

small and big universes and misses most of the

stars

One can use the volume weighted measure

parametrized by time t proportional to the scale

factor a. In this case the two parts of the

universe grow at the same rate, but the

non-inflationary one stops growing and collapses

while the big one continues to grow. Thus the

comparison of the volumes at equal times fails.

Scale factor cutoff, t a

When we make a cut, in the beginning inflation

does not provide us any benefit No gain in

volume. This could suggest that the growth of

volume during inflation does not have any

anthropic significance

But when we move the cut-off higher, the

comparison between the two parts of the universe

becomes impossible. The small part of the

universe dies early, whereas the inflationary

universe continues to grow. When we remove the

cut-off, we find the usual result Most of the

observers live in the universe produced by

inflation.

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Boltzmann Brains are coming!!!

BB1

BB3

Hopefully, normal brains are created even faster,

due to eternal inflation

Consider first the scale factor

cutoff, t a.

If the dominant vacuum cannot produce Boltzmann

brains, and our vacuum decays before BBs are

produced we will not have any problems with them.

Freivigel, Bousso et al, in preparation De

Simone, Guth, Linde, Noorbala, Salem,Vilenkin, in

preparation

Can we realize this possibility? Recall that

Ceresole, DallAgata, Giryavets, Kallosh, A.L.,

2006

The long-living vacuum tend to be the ones with

an (almost) unbroken supersymmetry,

. But people like us cannot live in a

supersymmetric universe.

In other words, Boltzmann brains born in the

stable vacua tend to be brain-dead.

More on this in the talks by Vilenkin and

Freivogel

Problems with probabilities

V

3

4

2

1

5

Time can be measured in the number of

oscillations ( ) or in the number of

e-foldings of inflation ( ). The

universe expands as

is the growth of volume during

inflation

Unfortunately, the result depends on the time

parametrization.

t21

t45

t 0

We should compare the trees of bubbles not at

the time when the trees were seeded, but at the

time when the bubbles appear

A possible solution of this problem

If we want to compare apples to apples, instead

of the trunks of the trees, we need to reset the

time to the moment when the stationary regime of

exponential growth begins. In this case we obtain

the gauge-invariant result

As expected, the probability is proportional to

the rate of tunneling and to the growth of volume

during inflation.

A.L., arXiv0705.1160

What if instead of the minimum at the top, we

have a flat maximum, as in new inflation?

Boundary of eternal inflation

A preliminary answer (Winitzki, Vanchurin, A.L.,

in progress) In the limit of small V, when the

size of the area of eternal inflation becomes

sufficiently small, the results, in the leading

approximation, do not depend on time

parametrization.

In general, according to the stationary measure,

if we have two possible outcomes of a process

starting at t 0

where ti is the time when the stationarity regime

for the corresponding process is established. The

more probable is the trajectory, the longer it

takes to reach stationarity, the better. For

example, the ratio of the probabilities for

different temperatures in the domains of the same

type is

Slight preference for lower temperatures, no

youngness paradox.

A.L. 2007

Moreover, we believe that the youngness paradox

does not appear even if one takes into account

inhomogeneities of temperature.

A.L., Vanchurin, Winitzki, in preparation

These results agree with the expectation that

the probability to be born in a part of the

universe which experienced inflation can be very

large, because of the exponential growth of

volume during the slow-roll inflation.

No Boltzmann Brainer

A.L., Vanchurin, Winitzki, in preparation

Stationary measure does not lead to the Boltzmann

brain problem

The ratio of BBs to OOs is proportional to the

ratios of the volumes of the universe when the

stationarity is reached for BBs and OOs (which

rewards OOs), multiplied by the extremely small

probability of the BB production in the

vacuum. Example In the no-delay situation

(A.L. 2006)

In a more general and realistic situation, with

the time delay, taking into account thermal

fluctuations, the result is very similar, no BBs

Conclusions

There is an ongoing progress in implementing

inflation in supergravity and string theory.

As on now, we are unaware of any non-inflationary

alternatives which are verifiably consistent.

CMB can help us to test string theory. If

inflationary tensor modes are discovered, we may

need to develop phenomenological models with

superheavy gravitino.

Looking forward, we must either propose something

better than inflation and string theory in its

present form, or learn how to make probabilistic

predictions based on eternal inflation and the

string landscape scenario. Several promising

probability measures were proposed, including the

stationary measure.

Bousso, Freivogel and Yang, 2007

Here our conclusions differ from those of

They confirmed that in the absence of

perturbations of temperature T, the probability

distribution to be born in the universe with a

given T depends on T smoothly, and does not

suffer from the youngness problem. However, when

they took into account perturbations of

temperature, they found, in the approximation

which they proposed, that

Here 1010 stays for the inverse square of the

amplitude of perturbations of temperature induced

by inflationary perturbations. The first term is

much greater, which leads to oldness paradox

Small T are exponentially better. Note, however,

that if one takes the limit when the amplitude of

perturbations vanishes, the coefficient in front

of the second term in the exponent blows up, and

the probability distribution becomes singular,

concentrated at T 3.3 K. This contradicts their

own result that in the absence of perturbations

of temperature, the probability distribution is

smooth.

A toy landscape model

Mahdiyar Noorbala, A.L. 2008

As an example, consider Bousso measure, assuming

first that Boltzmann brains can be born in all

vacua

However, stable vacua are not really stable. In

a typical situation in stringy landscape one

expects their decay rate

Ceresole, DallAgata, Giryavets, Kallosh, A.L.,

2006

Such vacua could be BB safe. If there are other

vacua, with a small SUSY breaking, they may be

stable but it is dangerous only if

Another advantage of the stationary measure

A.L., Vanchurin, Winitzki, in preparation

1. It does not suffer from the youngness paradox

Usually youngness problem appears because of the

reward of the delay of inflation by a factor of

But in the stationary measure, this reward is

taken back when taking into account time delay

for the onset of stationarity.

2. It does not lead to the Boltzmann brain problem

The ratio of BBs to OOs is proportional to the

ratios of the volumes of the universe when the

stationarity is reached for BBs and OOs (which

rewards OOs), multiplied by the extremely small

probability of the BB production in the vacuum.

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